Vitali convergence theorem
Let be -integrable functions on some measure space, for .
the sequence converges to in measure;
for every , there exists a set of finite measure, such that for all .
This theorem can be used as a replacement for the more well-known dominated convergence theorem, when a dominating cannot be found for the functions to be integrated. (If this theorem is known, the dominated convergence theorem can be derived as a special case.)
In probability , the definition of “uniform integrability” is slightly different from its definition in general measure theory; either definition may be used in the statement of this theorem.
- 1 Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications, second ed. Wiley-Interscience, 1999.
- 2 Jeffrey S. Rosenthal. A First Look at Rigorous Probability Theory. World Scientific, 2003.
|Title||Vitali convergence theorem|
|Date of creation||2013-03-22 16:17:10|
|Last modified on||2013-03-22 16:17:10|
|Last modified by||stevecheng (10074)|
|Synonym||uniform-integrability convergence theorem|